Before spooning yogurt out of the carton, pour off any liquid on top. If you use a particularly thick yogurt, you may need to thin the sauce with a little water.
While grill heats, whisk together yogurt, minced mint, lemon juice, and salt to taste. Mince 1 half of garlic clove and whisk into yogurt sauce.
03Mix lamb, parsley, onion, salt, pepper, and allspice with your hands in a bowl until just combined (do not overwork mixture or patties will be tough). Form into 4 (4 1/2-inch) patties (1/2 inch thick).
04Brush both sides of bread slices with 2 tablespoons oil (total for all slices) and grill, covered only if using a gas grill, turning over once, until golden, 1 to 2 minutes total. Rub 1 side of each toast with cut side of remaining garlic and season with salt.
05Grill patties on lightly oiled grill rack, covered only if using a gas grill, turning over once, until browned but still slightly pink in center, 5 to 7 minutes total.
06Toss together mesclun, whole mint leaves, remaining teaspoon oil, and salt and pepper to taste. Divide lamb among toasts on plates, then spoon sauce over and top with greens.